Problem: $\dfrac{ 9m + 8n }{ 8 } = \dfrac{ 2m + 2p }{ -9 }$ Solve for $m$.
Explanation: Multiply both sides by the left denominator. $\dfrac{ 9m + 8n }{ {8} } = \dfrac{ 2m + 2p }{ -9 }$ ${8} \cdot \dfrac{ 9m + 8n }{ {8} } = {8} \cdot \dfrac{ 2m + 2p }{ -9 }$ $9m + 8n = {8} \cdot \dfrac { 2m + 2p }{ -9 }$ Multiply both sides by the right denominator. $9m + 8n = 8 \cdot \dfrac{ 2m + 2p }{ -{9} }$ $-{9} \cdot \left( 9m + 8n \right) = -{9} \cdot 8 \cdot \dfrac{ 2m + 2p }{ -{9} }$ $-{9} \cdot \left( 9m + 8n \right) = 8 \cdot \left( 2m + 2p \right)$ Distribute both sides $-{9} \cdot \left( 9m + 8n \right) = {8} \cdot \left( 2m + 2p \right)$ $-{81}m - {72}n = {16}m + {16}p$ Combine $m$ terms on the left. $-{81m} - 72n = {16m} + 16p$ $-{97m} - 72n = 16p$ Move the $n$ term to the right. $-97m - {72n} = 16p$ $-97m = 16p + {72n}$ Isolate $m$ by dividing both sides by its coefficient. $-{97}m = 16p + 72n$ $m = \dfrac{ 16p + 72n }{ -{97} }$ Swap signs so the denominator isn't negative. $m = \dfrac{ -{16}p - {72}n }{ {97} }$